Dudley's martingale representation theorem states that if $W=\{W_t,\mathcal{F}_t;0\le t<+\infty\}$ is a standard one-dimensional Brownian motion, $0<T<+\infty$ and $\xi$ is $\mathcal{F}_T$-measurable, then there exists a progressively measurable process $Y=\{Y_t,\mathcal{F}_t;0\le t\le T\}$ satisfying $$\int_0^TY_t^2\mathrm{d}t<+\infty\text{ a.s.}$$ such that $$\xi=\int_0^TY_t\mathrm{d}W_t\text{ a.s.}$$ But this theorem does NOT have uniqueness. We need to find a progressively measureable process $Y$ such that $$0<\int_0^1Y_t^2\mathrm{d}t<+\infty\text{ a.s.,}\quad\text{but }\int_0^1Y_t\mathrm{d}W_t=0\text{ a.s.}$$

Could anyone give such an example?